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Question

Jean puts N identical cubes, the sides of which are 1 inch long, inside a rectangular box, each side of which is longer than 1 inch, such that the box is completely filled with no gaps and no cubes left over. What is N?

(1) 56 < N < 63
(2) N is a multiple of 3.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer

To be able to put N cubes into a rectangular box with no gaps and no left-over cubes, you must have the following equality: N = length × width × height. Moreover, if the length, width, and height are all greater than 1, then it must be true that N is the product of at least 3 primes. If N is itself prime or the product of just 2 primes (unique or not), then the condition fails.

So you can rephrase the question this way: is N the product of at least 3 primes?

Statement 1: SUFFICIENT. Since N must be an integer (it counts a set of objects), you can rephrase the statement: N is either 57, 58, 59, 60, 61, or 62. Now test each of these numbers.
57 = 3 × 19 = product of just 2 primes, so 57 is out.
58 = 2 × 29 = product of just 2 primes, so 58 is out.
59 is itself prime, so 59 is out.
60 = × 3 × 5 = product of 4 primes, so 60 can work. For instance, the dimensions of the box could be 4 by 3 by 5, or various other combinations of the primes.
61 is prime, so 61 is out.
62 = 2 × 31 = product of just 2 primes, so 62 is out.
Thus, the statement tells us that N must be 60.

Statement 2: NOT SUFFICIENT. It’s easy to come up with multiple examples that fit the conditions. For instance, we can reuse N = 60 (just be careful whenever you reuse a case from one statement in another statement), but then try N = 27 (= 3 × 3 × 3), which also fits. Many multiples of 3 can be legal values of N, so this statement is not enough.

The correct answer is A.

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